Regularity of linear waves at the Cauchy horizon of black hole spacetimes

Transcription

1 Regularity of linear waves at the Cauchy horizon of black hole spacetimes Peter Hintz joint with András Vasy Luminy April 29, 2016

2 Cauchy horizon of charged black holes (subextremal) Reissner-Nordström-de Sitter spacetime solution of Einstein-Maxwell system, black hole mass M > 0, charge Q > 0,

3 Cauchy horizon of charged black holes (subextremal) Reissner-Nordström-de Sitter spacetime solution of Einstein-Maxwell system, black hole mass M > 0, charge Q > 0, cosmological constant Λ > 0,

4 Cauchy horizon of charged black holes (subextremal) Reissner-Nordström-de Sitter spacetime solution of Einstein-Maxwell system, black hole mass M > 0, charge Q > 0, cosmological constant Λ > 0, topology: R t (0, ) r S 2, metric: g = µ(r) dt 2 µ(r) 1 dr 2 r 2 dσ 2 ; t = t F (r).

5 Penrose diagram:

6 Penrose diagram: Cauchy horizon CH + : boundary of domain of uniqueness of solution u to wave equation g u = 0 with Cauchy data on H I

7 Blue-shift effect and strong cosmic censorship (Simpson Penrose 73.) Observer A crosses CH + in finite time.

8 Blue-shift effect and strong cosmic censorship (Simpson Penrose 73.) Observer A crosses CH + in finite time. Observer B lives forever.

9 Conjecture (Penrose s Strong Cosmic Censorship) The maximal globally hyperbolic development of generic initial data for Einstein s field equations is inextendible as a suitably regular Lorentzian manifold.

10 Conjecture (Penrose s Strong Cosmic Censorship) The maximal globally hyperbolic development of generic initial data for Einstein s field equations is inextendible as a suitably regular Lorentzian manifold. Related work: Christodoulou (..., 99, 08), Dafermos ( 03, 05, 13), ongoing work by Dafermos Luk, Luk Oh.

11 Toy model: linear wave equation Cauchy problem for g u = 0 on Reissner Nordström.

12 Toy model: linear wave equation Cauchy problem for g u = 0 on Reissner Nordström. Theorem (Franzen, 14) For C initial data, u remains uniformly bounded near CH +.

13 Toy model: linear wave equation Cauchy problem for g u = 0 on Reissner Nordström. Theorem (Franzen, 14) For C initial data, u remains uniformly bounded near CH +. Theorem (Luk Oh, 15) For generic C initial data, u is not in Hloc 1 near any point of CH+.

14 Theorem (H. Vasy, 15) For C initial data on Reissner Nordström de Sitter, u solving g u = 0 has a partial asymptotic expansion near i +, u = u 0 +u, u 0 C, u (t ) e αt, where α > 0 depends only on the spacetime.

15 Theorem (H. Vasy, 15) For C initial data on Reissner Nordström de Sitter, u solving g u = 0 has a partial asymptotic expansion near i +, u = u 0 +u, u 0 C, u (t ) e αt, where α > 0 depends only on the spacetime. More precisely, u and its derivatives tangential to CH + lie in e αt H 1/2+α/κ 0.

16 Theorem (H. Vasy, 15) For C initial data on Reissner Nordström de Sitter, u solving g u = 0 has a partial asymptotic expansion near i +, u = u 0 +u, u 0 C, u (t ) e αt, where α > 0 depends only on the spacetime. More precisely, u and its derivatives tangential to CH + lie in e αt H 1/2+α/κ 0. κ > 0: surface gravity of the Cauchy horizon

17 Previous work Microlocal analysis/scattering theory approach: Melrose ( 93) Sá Barreto Zworski ( 97), Bony Häfner ( 08) Melrose Sá Barreto Vasy ( 14) Wunsch Zworski ( 11), Dyatlov ( 11 15) Vasy ( 13), H. Vasy ( 13)

18 Previous work Microlocal analysis/scattering theory approach: Melrose ( 93) Sá Barreto Zworski ( 97), Bony Häfner ( 08) Melrose Sá Barreto Vasy ( 14) Wunsch Zworski ( 11), Dyatlov ( 11 15) Vasy ( 13), H. Vasy ( 13) Energy estimates: Dafermos Shlapentokh-Rothman ( 15) Luk Sbierski ( 15)

19 Analysis near the exterior region Given u e lt H s 0, vanishing near H I, study regularity and decay of u. g u = f C c,

20 Analysis near the exterior region Given u e lt H s 0, vanishing near H I, g u = f C c, study regularity and decay of u. (E.g. l 0, s 0 = 0.)

21 Analysis near the exterior region Given u e lt H s 0, vanishing near H I, g u = f Cc, study regularity and decay of u. (E.g. l 0, s 0 = 0.) u is C. Quantitative bounds as t?

22 Microlocal red-shift effect g u = f C c, u smooth.

23 Microlocal red-shift effect g u = f C c, u smooth. Partial compactification suited for global analysis: M := [0, ) τ (0, ) r S 2, where τ = e t.

24 Microlocal red-shift effect g u = f C c, u smooth. Partial compactification suited for global analysis: M := [0, ) τ (0, ) r S 2, where τ = e t. Propagation of singularities on uniform version b T M of the cotangent bundle down to τ = 0. R: saddle point for the null-geodesic flow lifted to b T M.

25 Microlocal red-shift effect g u = f C c, u smooth. Partial compactification suited for global analysis: M := [0, ) τ (0, ) r S 2, where τ = e t. Propagation of singularities on uniform version b T M of the cotangent bundle down to τ = 0. R: saddle point for the null-geodesic flow lifted to b T M. If u e lt H s 0 near R, and s 0 > 1/2 + l/κ, then u e lt H near R.

26 Resonance expansions Spectral family g (σ) = e it σ g e it σ.

27 Resonance expansions Spectral family g (σ) = e it σ g e it σ. Meromorphic continuation and quantitative bounds for g (σ) 1 (Mazzeo Melrose 87, Guillarmou 04, Vasy 13, Wunsch Zworski 11, Dyatlov 11 15). Poles: resonances.

28 Resonance expansions Spectral family g (σ) = e it σ g e it σ. Meromorphic continuation and quantitative bounds for g (σ) 1 (Mazzeo Melrose 87, Guillarmou 04, Vasy 13, Wunsch Zworski 11, Dyatlov 11 15). Poles: resonances.

29 Resonance expansions Spectral family g (σ) = e it σ g e it σ. Meromorphic continuation and quantitative bounds for g (σ) 1 (Mazzeo Melrose 87, Guillarmou 04, Vasy 13, Wunsch Zworski 11, Dyatlov 11 15). Poles: resonances. Obtain: u = u 0 + u, u 0 C, u e αt H.

30 This gives asymptotics and decay in r r CH + + ɛ, ɛ > 0.

31 Analysis near the Cauchy horizon Need to work in domain containing CH +. Choice of extension does not affect waves in r > r CH +!

32 Analysis near the Cauchy horizon Need to work in domain containing CH +. Choice of extension does not affect waves in r > r CH +! Modify spacetime beyond CH + : Add artificial exterior region.

33 Analysis near the Cauchy horizon Need to work in domain containing CH +. Choice of extension does not affect waves in r > r CH +! Modify spacetime beyond CH + : Add artificial exterior region. g = µ dt 2 µ 1 dr 2 r 2 dσ 2.

34 Setup for the extended problem Study forcing problem g u = f, with u = 0 near H I H I,a.

35 Setup for the extended problem Study forcing problem g u = f, with u = 0 near H I H I,a. (Add complex absorbing potential Q Ψ 2 b beyond CH+ to hide additional trapping and H a. Study g iq.)

36 Microlocal blue-shift effect g u = f C c. Work near CH +. Recall τ = e t.

37 Microlocal blue-shift effect g u = f C c. Work near CH +. Recall τ = e t.

38 Microlocal blue-shift effect g u = f C c. Work near CH +. Recall τ = e t. If u e lt H near R, and u e lt H s in a punctured neighborhood of R within {τ = 0}, then u e lt H s near R, provided s < 1/2 + l/κ.

39 Microlocal analysis of the extended problem Green arrows: Future directed timelike vectors.

40 Microlocal analysis of the extended problem Green arrows: Future directed timelike vectors. Control u solving g u = f using standard hyperbolic theory near dashed surfaces, and microlocal elliptic regularity and propagation of singularities.

41 Obtain u e lt H s g u e lt H s 1 + u e lt H s 0, where

42 Obtain u e lt H s g u e lt H s 1 + u e lt H s 0, where s > s 0 > 1/2 + l/κ at H +, H +,

43 Obtain u e lt H s g u e lt H s 1 + u e lt H s 0, where s > s 0 > 1/2 + l/κ at H +, H +, s < 1/2 + l/κ at CH +. (s is a variable order function, and l < 0.)

44 Obtain u e lt H s g u e lt H s 1 + u e lt H s 0, where s > s 0 > 1/2 + l/κ at H +, H +, s < 1/2 + l/κ at CH +. (s is a variable order function, and l < 0.) If there are no resonances σ with Im σ = l, have better error term u e (l 1)t H s 0

45 Obtain u e lt H s g u e lt H s 1 + u e lt H s 0, where s > s 0 > 1/2 + l/κ at H +, H +, s < 1/2 + l/κ at CH +. (s is a variable order function, and l < 0.) If there are no resonances σ with Im σ = l, have better error term u e (l 1)t H s 0 (non-elliptic) Fredholm problem!

46 Obtain u e lt H s g u e lt H s 1 + u e lt H s 0, where s > s 0 > 1/2 + l/κ at H +, H +, s < 1/2 + l/κ at CH +. (s is a variable order function, and l < 0.) If there are no resonances σ with Im σ = l, have better error term u e (l 1)t H s 0 (non-elliptic) Fredholm problem! Get solvability of extended problem g u = f, and control of u near CH +.

47 Resonance expansion Solution u of g u = f has partial expansion u r>rch + = u 0 + u, u 0 C, u e αt H 1/2+α/κ 0 near CH +.

48 Resonance expansion Solution u of g u = f has partial expansion u r>rch + = u 0 + u, u 0 C, u e αt H 1/2+α/κ 0 near CH +. α > 0: spectral gap of g.

49 Resonance expansion Solution u of g u = f has partial expansion u r>rch + = u 0 + u, u 0 C, u e αt H 1/2+α/κ 0 near CH +. α > 0: spectral gap of g. u (t, r, ω) is smooth in t, ω. (Haber-Vasy 13.)

50 Resonance expansion Solution u of g u = f has partial expansion u r>rch + = u 0 + u, u 0 C, u e αt H 1/2+α/κ 0 near CH +. α > 0: spectral gap of g. u (t, r, ω) is smooth in t, ω. (Haber-Vasy 13.) H 1/2+0 (R r ) L (R r ) yields u (t ) e αt.

51 Other settings Our methods apply directly to subextremal Kerr-de Sitter black holes, small angular momentum a 0,

52 Other settings Our methods apply directly to subextremal Kerr-de Sitter black holes, small angular momentum a 0, subextremal Kerr-Newman-de Sitter black holes, charge Q, small angular momentum a 0.

53 Other settings Our methods apply directly to subextremal Kerr-de Sitter black holes, small angular momentum a 0, subextremal Kerr-Newman-de Sitter black holes, charge Q, small angular momentum a 0. differential forms (Maxwell)

54 Other settings Our methods apply directly to subextremal Kerr-de Sitter black holes, small angular momentum a 0, subextremal Kerr-Newman-de Sitter black holes, charge Q, small angular momentum a 0. differential forms (Maxwell) With some additional work: symmetric 2-tensors (linearized gravity)

55 Other settings Our methods apply directly to subextremal Kerr-de Sitter black holes, small angular momentum a 0, subextremal Kerr-Newman-de Sitter black holes, charge Q, small angular momentum a 0. differential forms (Maxwell) With some additional work: symmetric 2-tensors (linearized gravity) Only potential issue for large a: resonances in Im σ 0 ( mode stability, see Whiting 89, Shlapentokh-Rothman 14 for Kerr)

56 Outlook Shallow resonances. Mode solution g (e iσt v(x)) = 0 has v H 1/2 Im σ/κ 0 at CH + ; could be C in principle. Study location and regularity properties of shallow resonances.

57 Outlook Shallow resonances. Mode solution g (e iσt v(x)) = 0 has v H 1/2 Im σ/κ 0 at CH + ; could be C in principle. Study location and regularity properties of shallow resonances. Λ = 0. Localize analysis near black hole region; in the exterior, polynomial decay according to Price s law (Tataru 13, Metcalfe Tataru Tohaneanu 12). Get polynomial decay near CH +. (Luk Sbierski 15, H. 15)

58 Outlook Shallow resonances. Mode solution g (e iσt v(x)) = 0 has v H 1/2 Im σ/κ 0 at CH + ; could be C in principle. Study location and regularity properties of shallow resonances. Λ = 0. Localize analysis near black hole region; in the exterior, polynomial decay according to Price s law (Tataru 13, Metcalfe Tataru Tohaneanu 12). Get polynomial decay near CH +. (Luk Sbierski 15, H. 15) Nonlinear problems. Einstein s field equations. Work in progress by Luk Rodnianski, Dafermos Luk, Luk Oh.